So that's it for stability review. What we're going to do now so we're going to go back to dynamics. And we're going to pick up for 50/10 somewhat left off you derived equations, rigid body did all the feedback control with MRPs and again that stuff you can reuse here a lot. But our equations of motion are gonnaget much more complicated I think in 50/10 we show you the stuff but you didn't actually derive anymore the full things. Here, you will derive them so we're going to go through these steps how to put them together because this is very much the details. If all the details, each individual step will not seem hard when you put it all together one little issue or sometimes it's so much details, you just lose track of the forest, you just see all the trees, right? It's easy to get lost you have to find some algebraic simplifications. I do recommend in this homework to I'm okay with you turning in mathematica code and how to derive this if you can do that. Fantastic or you might use things like Math Lab symbolic manipulators too. Help me with some of these algebra and sines and cosine and did I miss some term perfectly fine. I'm not trying to teach you algebra and trig here. I'm assuming you can do that as a PhD student or grad student here, it's really about can you put this stuff together in a meaningful way. So there's a lot of this stuff that's right by hand and double check your algebra that you're doing. And we'll talk about tips and tricks and how to actually double check complicated algebra as you're developing it. Right, so we'll be deriving this now. So the equations of motion, the truth of gyroscopic, what's the truth of gyroscopic, basically, like I'm teaching undergraduates the same stuff right now and then asked me so how do I visualize this term? And my answer is don't try, I'm just going to give yourself headaches. Anything more than simple two frames and things are twisting even after 25, 30 years of doing this I still go, let me quickly use transport theorem and derive this. And then I see exactly what's going to be in there because it's often not intuitive, we're not used to thinking and twisting and especially not with multiple frames and we'll have quite a few frames, lots of wheels, what happens? So we want to have rigorous use of rotating frames, physical principles how to put this together and then learn how to interpret. That's what this class will be looking at, all the stuff. Okay, so the truth of the gyroscopic sizes, trust your math. So let's talk about this, we are going to be developing the equations of motion, a rigid hub and then we're adding for now a single variable speed, control moment gyroscope. So what does the classic control moment gyroscope do? Abby, do you remember? >> [INAUDIBLE] >> Right, so it's an actual it's an attitude actuacion device, right? Jess, do you remember how, what makes the CMG different than just a reaction wheel or a flywheel, right. [INAUDIBLE] >> Almost you said fixed axis and fixed speed, it's not fixed speed the reaction will has a fixed rotation axis. So we'll be calling that GS and you can spin it, but then you apply a torque and so as you spin up the wheel, you're applying one newton meter onto the wheel that means the wheel is applying minus one newton meter onto you. And you will have a one to one relationship for every newton meter you pit in, you get one newton meter back onto yourself and while the wheel starts accelerating this way, you will start rotating this way, that's what your control says, rotate to this direction. So the wheel speed is not constant on the wheel versus the CMG that's where the wheel speed nominally is constant, not at launch, at launch it will be zero then a motor, there's a motor here that actually spins it up on the CMG as well. And then after it's spinning at 10,000 rpm or whatever speeds it's going to they try to go to a pretty high speed. Then that motor just keeps that wheel speed reasonably constant. It's not super accurate, it doesn't have to be that precise but as long as we know, the wheel speeds were good but it tries to keep it normally constant. Cause otherwise, as you rotate your spacecraft and do stuff, you would naturally dump momentum into and pull it out of the wheel and we want the wheel to have a fixed speed, right? So that only happens with the serval loop. And that's one of the things you will actually learn how to develop we'll see that in all our math. So we will spin it up so like a flywheel, we still have a motor torque here spinning up the wheel and now we try to hold it spinning. So this is my momentum axis, right fixed of the wheel relative to the body and you're right, the spin axis starts to vary. So the way the spin axis varies is there's a frame that kind of holds this wheel in place, clamps it in. It may not wrap on both sides, maybe just the one armed frame, but something holds that thing in place. And here I did it on both sides because then I can assume the center of mass of this frame is still the center of mass of the wheel. If it's imbalanced it gets much, much nastier that's somebody else's thesis recently, they derived all of that. So that's what I'm assuming here and now the gimbal rate gamma dot, that's what's so gimbaling is called if this wheel is spinning and that's my spin axis. I'm going to slowly twist this wheel spin axis and you've probably done these experiments with flywheels, you hold the wheel, right? Some physics class or dynamic class and then you twist the wheel and all of a sudden it really it kind of you get this big reaction and the faster the wheel spins, the stronger that torque is. And the reason for that comes down to h dot equal to L with these with the gyroscopes, that momentum, if the momentum of that wheel and now I twisted 10 degrees the change in inertial momentum is equal to the torque applied to that wheel. So that change in momentum is the vector going from here to here, that little vector right here, right? And that is orthogonal, however rotated, that's the axis you're going to torque this way to get it to move like that. So that's why we're going to torque this to get this access to twist and it creates this big gyroscopic torque. So, as we derived the equation at some point, I will point out this math and you will see a term that is the big wheels, the wheel speed which is omega as the wheel speed relative to the gimbal frame and it's a function of gamma dot. And so that gyroscopic torque, if you gimbal it, it gets this huge torque and it will be about the GT axis, the third axis. So it gets complicated, I'm spinning about GS and gambling about GG and the output will be well actually about all the access but the big torque will be about GT. But there's other ones, there's many equations of emotions, you will find in a lot of CMG papers that just kind of going to, let's look at the big one, everything else we ignore. And often they don't even mention why or how, but they just ignore it because it is the big one and they can do it. But if you do that, you cannot look at work, energy principles, you cannot look at any momentum conservation. None of those things will validate your code because you're dropping terms. So what we're doing in this class is what I developed for these variable speed CMGs which is a complete physics based model of a balanced system granted. But we've done imbalanced in research and so this will actually satisfy energy momentum, work, energy principles, all those kinds of things. Which will be great because this will be a tool that you will use in checking your homework to make sure the simulation is all correct and that you haven't missed the term somewhere or flip the sign or something like that, right? So big omega wheel speed little gamma dot is our gimbal rate and that's what we'll be doing and that's a rate, not an acceleration and that's what we're looking for. So why do we use these CMGIs? Normally they're constant and that's good then we can gimbal them around. But you are now, with wheels, the GS axes is what determines how you're getting a torque. If I'm spinning about my 1,0,0 axis my first one, right the wheel and then I'm torqing it about that one Newton meter I'm going to have minus one Newton meter back onto my body. But the torque I get for accelerating the first wheel is always going to be about my first body access because that's how I lined it up. Whereas with the CMG, because I am twisting those axes, GS and GT are not constant as seen by the body. GG typically is that's how you bolted down the CMG device, but the GS GT were actually twisting and all of a sudden we're going to have time varying terms in our inertia mass matrix. And our actuacion axis aren't just fixed in the body, they're actually changing with time. So CMGs will have a problem with, well, all momentum based attitude control devices have an issue with saturation at some point. Either the wheel speed accelerates to a point of, I'm about to fall apart right there. Say, okay, you've got the 6000 RPMs beyond that things break down, not good, right? With CMGs were not accelerating the wheels, but we're rotating those axes. So if you apply a constant torque, I'll show the math later what happens is all these axes begin to become co-planer and the torque have to produce is orthogonal to this plane, that's called a gimbal lock. You may have heard of the term gimbal lock situation. So then we have singularities where all of a sudden but I want to torque here and you go yeah you're not going to get it but I've got 15 CMGs, I spent $6 billion on these things and yet but you're not going to get it. It just doesn't matter you will have at some point you keep applying an external torque, there's a singularity there as well, right? So all these reaction meals have that or CMG devices go ahead. So if you're just looking at this for a moment and respect and we separate the hub and the wheel, a reaction wheel can both move your hubs momentum vector. And also scale it. >> Yes. >> Changing momentum but the classic CMGs only just moved the hubs momentum vector, it can't skip. >> Well no you there is you can reconfigure it where you moved that momentum out of you have multiple wheels right now we're talking about one if I had one yes. But if you have multiple wheels, the CMGs we actually typically line them up all its momentum vectors canceled sum up to zero. Otherwise you have a huge momentum bias which makes it super stable but stable typically means it resists departures from where you want to go. So it doesn't make it very maneuverable, right? That makes it tricky so you can cancel that with the net thing there's zero you can reconfigure it and all of a sudden you have now gone from having a slower momentum vector in the body to a very fast one. So you will have scaled it and then when you stop the gambling it's going to come back down, total momentum has to be preserved. So if you zero them out somehow you must have slowed down the body. Yeah so this this is an interesting thing. So that's CMGs, so why on earth are we doing variable speed CMGs? This is something this is one of my earlier papers I wrote. I saw Chris Hall and his student at the time, Kevin Ford he was an astronaut who was getting his PhD with Chris, they were talking about gimbled reaction wheels. I thought that was interesting because the math formulation was done in a way such that you get both effects all at once and it wasn't solved quite like this the way I'm doing. But I thought that was an interesting concept but really Salvador Dali actually not a professor Denim came up with the idea of calling them variable speed CMGs because it really we don't use them as reaction wheels nominally. There is a motor about the spin axis and you have to do it when you spin them up. But we really use the master control moment gyroscopes nominally but that has a constant wheel speed assumption and that creates singularities. Well if I can actually also create torques by spinning up and spinning down, you get to control modes out of one device. You can accelerate the wheel which is acting like a flywheel, a reaction wheel or I can gimbal the wheel and that acts like a CMG or I can do both at the same time. And now I can get a talk about GSM, the G T and if all the GTs lineup, there's ways to avoid that, all the GS is lined up and I can always get controllability by changing the wheel speed. So people have been looking at those and so augmenting CMGIs with wheels and making one wheel that can do both that's what we'll be talking about in great detail. There's benefits, there's a huge null space with variable speed CMGs, we typically have four of them, classic CMG configuration. It's a 3D control problem, so there's already a one dimensional null space. But if you have a variable speed CMGs, those four devices will give you eight dimensional control and it's a 3D control problems. So I have a five dimensional null space, there's many ways I can spin up this, gimbal this, gimbal and spin up the other one and not produce a torque because it all cancels. So we can reconfigure gimbal angles, we can reconfigure speeds of the wheels. That's kind of really cool actually and some of these, Surrey has actually flown some of them as well others are looking at them. The other application, the Air Force had a big project looking at variable speeds. CMGI's for attitude and energy storage devices because if I'm in the sun I usually took all the power and put in a battery, batteries don't scale well, especially to small sizes. It's a big honking chemical thing. And if we can just put all that power into the wheels, spin them up more and more and more and adjust the gimbals to dog producer torque. And then when you go through the shadow, use the dynamo effect, you start breaking these wheels and biased, you slow, not breaking as in breaking them but slowing them down as you used better words. To slow down the wheels, you can pull mechanical energy and extract in that braking process, electrical energy. The challenge there is the speeds aren't just 10,000 RPMs, they're talking 40,000, 50,0000, 60,000 RPMs which requires typically magnetic suspension. These are already a C while reaction will make cost this much a CMG will cost you 100 times more, it's much more expensive. So you need that big but you get way more for it. So you get huge torques. So if you have to move rapidly or you have to move very big stuff space stations, they actually use a double gambled CMG and it has very few. That's the only one I can think of right now that's flying a double gambled one. Has similar effects the biggest benefit I think for them is they can absorb torques longer before they have to dump momentum than a single gamble. But you will see the math, it's just like wow, okay where do we go? So with a variable speed, we can avoid some of these classic systems very redundant which will be interesting. We'll have lectures talking about small spaces and how to take advantage of it and how to maybe avoid gimbal locks completely by just tweaking the wheel speeds and how do we go with that? All right, that's where we'll be going with this and that's part of things you will simulate. And again, energy storage devices that some of the motivations for that. Paul. >> Still have the actively for locks [INAUDIBLE] >> Well, once you have eight control authority, they're non existent because you won't have those issues where you cannot produce the requested torque. The problem is if you move in the space station and you're taking that CMG and all of a sudden using it like a reaction wheel, the math says sure. The reality says, yeah, but you can be spinning all of a sudden not that a few 100 RPM or a few 1000 you might be tens of thousands all of a sudden because the torques required are huge. The CMGS you will see I put in a small input torque to twist it and we actually take advantage of that huge gyroscopic torque. So that faster that bicycle wheels spinning when you were holding it and you twist it was like whoa, you didn't put much effort into it and all of a sudden it's like a wild horse, right? That's what we're taking advantage of. That's why you're paying so many zeros for this device, small input, huge amplification on the output, whereas the reaction wheels, like I'm going to accelerate it. So you put in and you get back what you put in one to one, that's it and also you can see what the mathematics say, no singularity, the reality says saturation very quickly and those are the challenges you have to worry. So we will talk about that, we can do the classic one that will avoid singularity but it's not very practical because of saturation issues. So are there better ways to avoid it? And we'll look at singularity, avoidance strategies that used the variable speed in a subtle way, not in a brute force way. Yeah, so that's kind of my motivational slide. Now the other motivation is we could derive equations of motion using a bunch of reaction wheels or we could derive the equations of motion using a bunch of CMGS. But guess what if we derive the equations of motion using a variable speed CMG we've done everything at once, right? So you have the complex one that will use and study and analyze and it's actually realistic what a CMG does because every CMG has to spin up and you have to control that wheel speed. And you will have a physics based simulation that can emulate all of that and you can also just lock the gimbal rates and make it act like a reaction wheel as well. I know things will simplify greatly at that point, okay?
